The general expression of the ℓ = 2 straight helical heliotron field is obtained by integrating Biot-Savart's formula after expanding the integrand. Solving the differential equation of the line of force, a general expression for the magnetic surfaces, which is in good agreement with the results of a computer calculation up to the separatrix, is obtained. The rotational transform and the shear of this field are calculated as functions of the average radius of the magnetic surface. For a certain range of α*, the ratio of the longitudinal field produced by the Bz–coil to the longitudinal component of the field produced by the helical coil, the field cannot form a closed magnetic surface. The width of this range, the forbidden zone, is a function of γ = 2πa/p, where a is the radius of the helical winding and p is the pitch. For small γ, the zone is wide and the closed magnetic surface has very small shear. For large γ, the zone is narrow and the magnetic surface has very small rotational transform and shear. Only for intermediate γ, the zone is narrow and rotational transform and shear are both large. These values of γ could be considered to be optimum values for plasma confinement.
Effect of an external helical field on a rotating magnetic island